Digital noise-shaping filter with real coefficients and method for making the same

ABSTRACT

A digital noise-shaping filter with real coefficients for delta-sigma data converters used in a digital amplifier, 1-bit digital/analog converter, 1-bit analog/digital converter and the like, and a method for making the same. The present digital noise-shaping filter has improved noise suppression performance and system stability and reduced calculation complexity. To this end, the digital noise-shaping filter comprises a noise transfer function expressed by NTF(z)=−1+a 1 z −1 +a 2 z −2 +Λ+a N z −N . The noise transfer function has optimum real coefficients or real coefficients approximating them.

TECHNICAL FIELD

[0001] The present invention relates to a digital noise-shaping filterwith real coefficients for delta-sigma data converters used in a digitalamplifier, 1-bit digital/analog converter (DAC), 1-bit analog/digitalconverter (ADC) and the like, and a method for making the same.

BACKGROUND ART

[0002] The reasons for employing a conventional digital noise-shapingfilter and problems with the digital noise-shaping filter are asfollows.

[0003] 1. Reasons for Employing Noise-Shaping Filter

[0004] 1. Oversampling

[0005] Oversampling is typically used in a variety of technical fieldssuch as a delta-sigma data converter, digital amplifier, etc.Oversampling means sampling of the original signal at a higher samplingrate than a normal sampling rate at which the original signal can beheld and restored with no loss in information thereof. For example,assuming that an audio frequency band ranges from 20 Hz to 20 kHz, thensampling at the normal sampling rate signifies sampling of an audiosignal at a minimum of about 40 kHz on the basis of the Nyquist samplingtheorem. But, oversampling means sampling of the audio signal at ahigher sampling rate than the minimum sampling rate based on the Nyquistsampling theorem. For example, 8-times (×8) oversampling is defined assampling the above audio signal at a frequency of eight times theminimum sampling frequency, or on the order of 320 kHz (see FIG. 1).

[0006] In case oversampling is carried out as mentioned above, thesignal and quantization noise have spectra varying as shown in FIG. 2.In FIG. 2, the signal is indicated by a solid line and the quantizationnoise is indicated by a dotted line. The reason why the spectra of thesignal and quantization noise varies as shown in FIG. 2 is disclosed inmost teaching materials related to discrete signal processing (see: AlanV. Oppenheim, Ronald W. Schafer with John R. Buck, DISCRETE-TIME SIGNALPROCESSING 2nd ed. pp 201-213 (Prentice Hall Signal Processing Series,Upper Saddle River, N.J., 1999)).

[0007] It can be seen from FIG. 2 that a band that the signal occupieson the standard frequency axis owing to oversampling is reduced in widthcompared to that prior to oversampling in inverse proportion to anoversampling ratio. An appropriate low pass filter can be used to removea great portion of noise from the signal having such a reducedbandwidth. A noise-shaping filter can also be used to still furtherreduce the amount of noise energy of the band where the signal exists,by changing the shape of noise distribution by bands.

[0008]FIG. 3 shows spectra of the signal and quantization noise shapedby the noise-shaping filter, wherein the signal is indicated by a solidline, the quantization noise before being shaped is indicated by adotted, straight line, and the quantization noise after being shaped isindicated by a dotted, curved line. From comparison between thequantization noise before being shaped and the quantization noise afterbeing shaped in FIG. 3, it can be seen that the quantization noise ofthe signal band is significantly reduced in amount owing to the noiseshaping.

[0009] 1.2 Pulse Width Moldulation (PWM) and Requantization

[0010] PWM is one of methods for expressing a quantized signal (see FIG.4). In this PWM technique, each discrete signal has a fixed amplitude(an amplitude on the vertical axis in FIG. 4), which represents aphysical amount such as a voltage, and a pulse width on the time axis,which varies in proportion to the magnitude of the original signal. Anappropriate low pass filter can be used to restore the resultingmodulated signal with both the original signal and harmonic componentsinto the original signal. The PWM technique is used mostly in a digitalamplifier, delta-sigma converter, etc.

[0011] Signal modulation by the PWM technique necessitates a signalprocessor that has a higher degree of precision on the time axis than asampling frequency of the original signal, in that the magnitude of theoriginal signal is expressed not by a pulse amplitude, but by a pulsewidth. For example, for quantization of a signal sampled at 44.1 kHzinto a 16-bit signal, the signal processor is required to have aprocessing speed of 44.1 kHz×2¹⁶≈2.89 GHz. In some cases, a frequencythat is twice as high as the above processing speed may be used forquantization according to a given PWM mode.

[0012] Further, the PWM cannot help generating undesired harmoniccomponents due to its inherent characteristics. In this regard,oversampling must be carried out to reduce the harmonic components,resulting in the precision on the time axis becoming a higher frequencythan that in the above example. For example, for 8-times oversampling,the signal processor is required to have an operating frequency of about2.89 GHz×8≈23.12 GHz.

[0013] However, it is practically impossible to embody such a high-speedsignal processor. For this reason, the resolution of quantization mustbe lowered to a smaller value than the 16-bit value in the aboveexample, which is typically called requantization. A requantized signalhas a greater error compared to the original signal, which is expressedas noise components of the original signal. A noise-shaping filter isused to compensate for such an error.

[0014] 2. Noise-Shaping Filter

[0015] A noise-shaping filter functions to shape the spectrum ofquantization noise in a delta-sigma data converter. FIG. 5 shows thestructure of a conventional noise-shaping filter. In this conventionalnoise-shaping filter, a digital input signal {circumflex over (x)} of bbits is quantized into an output signal {circumflex over (x)}+e_(ns) ofb′ bits, where b′ is smaller than b. The component e_(ns) of the outputsignal is a noise component after the input signal is passed through theentire system. A component e_(rq) is a difference between a signalbefore being quantized and a signal after being quantized. Thenoise-shaping filter can shape the quantization noise by passing such asignal difference through an appropriate filter, feeding the resultingvalue back to the input signal and adding it to the input signal. Atransfer function of the appropriate filter is defined as H(z).

[0016] The noise-shaping filter acts to reduce noise components at aspecific frequency band in question by appropriately shaping thespectrum of quantization noise. A noise transfer function of thenoise-shaping filter of FIG. 5 can be defined as in the followingequation 1.

[0017] Equation 1 $\begin{matrix}{{N\quad T\quad {F(z)}} \equiv \frac{E_{n\quad s}(z)}{E_{r\quad q}(z)}} & \left\lbrack {{Equation}\quad 1} \right\rbrack\end{matrix}$

[0018] where, E_(ns)(z) and E_(rq)(z) are z-transforms of e_(ns) ande_(rq), respectively.

[0019] The noise transfer function can be derived from a conceptualdiagram of FIG. 5 as in the below equation 2.

NTF(z)=H(z)−1   Equation 2

[0020] The noise transfer function exerts an important effect on theperformance of a conventional noise-shaping filter. The conventionalnoise-shaping filter has a noise transfer function expressed by thefollowing equation 3.

NTF(z)=−(1−z ⁻¹)^(N tm Equation) 3

[0021] where, N is a natural number, which is an order of the filter.

[0022] For example, a noise transfer function of a second-order(order-2) filter can be obtained as in the below equation 4 by expandingthe noise transfer function of the above equation 3.

NTF(z)=−1+2z ⁻¹ −z ⁻²   Equation 4

[0023] Similarly, a noise transfer function of a third-order filter canbe expressed as in the following equation 5.

NTF(z)=−1+3z ⁻¹−3z ⁻² +z ⁻³   Equation 5

[0024] The below table 1 shows coefficients of respective terms in noisetransfer functions of second-order to seventh-order filters for 8-timesoversampling, where all constant terms are −1. TABLE 1 ORDER 2nd 3rd 4th5th 6th 7th a₁ 2 3 4 5 6 7 a₂ −1 −3 −6 −10 −15 −21 a₃ 1 4 10 20 35 a₄ −1−5 −15 −35 a₅ 1 6 21 a₆ −1 −7 a₇ 1

[0025] As seen from the above equation 4, equation 5 and table 1, thecoefficients of the noise transfer function of the conventionalnoise-shaping filter are integers (see: J. M. Goldberg, M. B. Sandler,Noise Shaping and Pulse-Width Modulation for an All-digital Audio PowerAmplifier, J. Audio Eng. Soc., vol. 39 pp. 449-460 (June 1991)).

[0026] The conventional noise-shaping filter is desirably simple toconstruct because the coefficients of the respective terms in the noisetransfer function thereof are simple integers, but has a disadvantage inthat it cannot maximize noise suppression performance.

[0027] Further, the higher order of the noise transfer function resultsin an instability in the entire system although it is able to improvethe performance of the conventional noise-shaping filter (see: Alan V.Oppenheim, Ronald W. Schafer with John R. Buck, DISCRETE-TIME SIGNALPROCESSING 2nd ed. pp 201-213 (Prentice Hall Signal Processing Series,Upper Saddle River, N.J., 1999)). That is, the order of the filtercannot unconditionally increase in consideration of the systemstability.

[0028] Accordingly, there is a need for a noise-shaping filter tominimize an instability of the entire system and use a noise transferfunction capable of maximizing noise suppression performance, moreparticularly coefficients of the noise transfer function, orcoefficients having values other than integers.

DISCLOSURE OF THE INVENTION

[0029] Therefore, the present invention has been made in view of theabove problems, and it is an object of the present invention to providea digital noise-shaping filter and a method for making the same, whereinrespective coefficients of a noise transfer function of thenoise-shaping filter are not simple integers, but appropriate realcoefficients.

[0030] It is another object of the present invention to provide adigital noise-shaping filter having real coefficients capable ofmaximizing noise suppression performance, and a method for obtaining thereal coefficients of the noise-shaping filter.

[0031] It is a further object of the present invention to provide adigital noise-shaping filter with real coefficients which showsexcellent noise suppression performance over those of conventionalnoise-shaping filters in the same order, and a method for making thesame.

[0032] It is another object of the present invention to provide adigital noise-shaping filter with real coefficients which is operable ina lower order than those of conventional noise-shaping filters to obtainthe same noise suppression performance as those of the conventionalnoise-shaping filters, resulting in an increase in system stability, anda method for making the same.

[0033] It is yet another object of the present invention to provide adigital noise-shaping filter which has real coefficients approximatingoptimum values so that it can have almost the same performance as theoptimum performance without increasing a calculation complexity, and amethod for making the same.

[0034] In accordance with one aspect of the present invention, the aboveand other objects can be accomplished by the provision of a digitalnoise-shaping filter for a delta-sigma data converter, comprising anoise transfer function expressed byNTF(z)=−1+a₁z⁻¹+a₂z⁻²+Λ+a_(N)z^(−N), the noise transfer function havingreal coefficients, where NTF(z) is a z-transform of the noise transferfunction, a₁, a₂, . . . , a_(N) are the real coefficients of the noisetransfer function, and N is an order of the noise-shaping filter.

[0035] Preferably, the real coefficients of the noise transfer functionmay be obtained by 1) defining an objective function enabling aquantitative evaluation of the performance of the noise-shaping filter;2) obtaining real coefficient conditions capable of optimizing theobjective function; and 3) mathematically calculating optimum realcoefficients satisfying the real coefficient conditions.

[0036] In accordance with another aspect of the present invention, thereis provided a method for making a digital noise-shaping filter for adelta-sigma data converter, the digital noise-shaping filter comprisinga noise transfer function expressed byNTF(z)=−1+a₁z⁻¹+a₂z⁻²+Λ+a_(N)z^(−N), the noise transfer function havingreal coefficients a₁, a₂, . . . , a_(N), wherein the real coefficientsof the noise transfer function are obtained by the steps of a) definingan objective function enabling a quantitative evaluation of theperformance of the noise-shaping filter; b) obtaining real coefficientconditions capable of optimizing the objective function; and c)mathematically calculating optimum real coefficients satisfying the realcoefficient conditions.

[0037] Preferably, the real coefficients of the noise transfer functionmay be values approximating the optimum real coefficients, and theapproximate real coefficients may be values approximated to four or moredecimal places by a binary number.

BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The above and other objects, features and other advantages of thepresent invention will be more clearly understood from the followingdetailed description taken in conjunction with the accompanyingdrawings, in which:

[0039]FIG. 1 is a conceptual diagram of general 8-times oversampling;

[0040]FIG. 2 is a conceptual diagram of 8-times oversampling in terms offrequency;

[0041]FIG. 3 is a conceptual diagram of noise shaping by a conventionalnoise-shaping filter;

[0042]FIG. 4 is a conceptual diagram of PWM;

[0043]FIG. 5 is a conceptual diagram of the conventional noise-shapingfilter;

[0044]FIG. 6 is a graph showing a noise shaping gain of a second-ordernoise-shaping filter in accordance with the present invention;

[0045]FIG. 7 is a graph showing a noise shaping gain of a third-ordernoise-shaping filter in accordance with the present invention;

[0046]FIG. 8 is a graph showing a noise shaping gain of a fourth-ordernoise-shaping filter in accordance with the present invention;

[0047]FIG. 9 is a graph showing a noise shaping gain of a fifth-ordernoise-shaping filter in accordance with the present invention;

[0048]FIG. 10 is a graph showing a noise shaping gain of a sixth-ordernoise-shaping filter in accordance with the present invention;

[0049]FIG. 11 is a graph showing a noise shaping gain of a seventh-ordernoise-shaping filter in accordance with the present invention;

[0050]FIG. 4is a graph showing an order-based noise shaping gain of adigital noise-shaping filter in accordance with the present invention;

[0051]FIG. 13 is a graph showing a noise shaping gain of a second-ordernoise-shaping filter whose real coefficients are approximated to onlyfour decimal places by a binary number;

[0052]FIG. 14 is a conceptual diagram of an apparatus for approximatingan optimum real coefficient of the first term of a second-ordernoise-shaping filter for 8-times oversampling;

[0053]FIG. 15 is a graph showing a noise shaping gain of a second-ordernoise-shaping filter whose real coefficients are approximated to twodecimal places by a binary number; and

[0054]FIG. 16 is a graph showing a noise shaping gain of a second-ordernoise-shaping filter whose real coefficients are approximated to threedecimal places by a binary number.

BEST MODE FOR CARRYING OUT THE INVENTION

[0055] 1. Expressions of Noise Transfer Function of DigitalNoise-Shaping Filter

[0056] A digital noise-shaping filter with real coefficients accordingto the present invention has a noise transfer function NTF(z) expressedas in the following equation 6.

[0057] Equation 6 $\begin{matrix}{{N\quad T\quad {F(z)}} = {{- 1} + {\sum\limits_{k = 1}^{N}{a_{k}z^{- k}}}}} & \left\lbrack {{Equation}\quad 6} \right\rbrack\end{matrix}$

[0058] where, N is an order of the noise transfer function and a_(k) isa coefficient of the noise transfer function.

[0059] In this invention, coefficients of the noise transfer functionare real coefficients. Expanding the right side of the above equation 6,the result is obtained as in the below equation 7.

NTF(z)=−1+a ₁ z ⁻¹ +a ₂z⁻² +Λ+a _(N)z^(−N)   Equation 7

[0060] 2. Optimization of Real coefficients of Noise Transfer Function

[0061] Real coefficients of a noise transfer function can be optimizedand approximated in the below manner.

[0062] 1) Define an objective function enabling a quantitativeevaluation of the performance of a noise-shaping filter.

[0063] 2) Obtain conditions of real coefficients capable of optimizingthe objective function.

[0064] 3) Obtain optimum real coefficients of the noise-shaping filterby mathematically calculating optimum real coefficients satisfying theabove conditions.

[0065] 1) Definition of Objective Function for Quantitative Evaluationof Performance of Noise-Shaping Filter

[0066] An objective function is defined as the ratio of the amount ofenergy of quantization noise before being shaped to the amount of energyof quantization noise after being shaped, at a frequency band ofinterest. A physical meaning of this definition is that a specific ratiorepresents a noise reduction degree.

[0067] The definition of the objective function is made in thisspecification for the first time. The objective function willhereinafter be referred to as a ‘noise shaping gain’, which can beexpressed as in the following equation 8.

[0068] Equation 8 $\begin{matrix}{{NSG} \equiv {10\log_{10}\frac{P_{r\quad q}}{P_{n\quad s}}}} & \left\lbrack {{Equation}\quad 8} \right\rbrack\end{matrix}$

[0069] where, NSG is a noise shaping gain, and P_(ns) and P_(rq) arepowers of noise components e_(ns)(n) and e_(rq)(n) in the above equation5, namely, powers in a signal band.

[0070] The noise components e_(ns)(n) and e_(rq)(n) are random signals.Power of each of the random signals can be defined as an expectationvalue of the square of each random signal. If a mean value of eachrandom signal, or noise component, is ‘0’, the expectation value of thesquare of each random signal is a variance. The powers of the noisecomponents can be calculated with respect to a minimum quantizationscale Δ as in the below equations 9 and 10, respectively.

[0071] Equation 9 $\begin{matrix}{P_{rq} = {{\xi \left\{ {e_{rq}^{2}\lbrack n\rbrack} \right\}} = {{\frac{1}{2\pi}{\int_{- \frac{\pi}{M}}^{\frac{\pi}{M}}{\sigma_{e}^{2}{\omega}}}} = {\frac{\sigma_{e}^{2}}{M} = \frac{\Delta^{2}}{12M}}}}} & \left\lbrack {{Equation}\quad 9} \right\rbrack\end{matrix}$

[0072] Equation 10 $\begin{matrix}{\begin{matrix}{P_{n\quad s} = \quad {{\xi \left\{ {e_{n\quad s}^{2}\lbrack n\rbrack} \right\}} = \left. {\frac{1}{2\pi}{\int_{- \frac{\pi}{M}}^{\frac{\pi}{M}}\sigma_{e}^{2}}} \middle| {{NTF}\left( ^{\omega} \right)} \middle| {}_{2}{\omega} \right.}} \\{= \quad \left. {\frac{\Delta^{2}}{12M}\int_{- \frac{\pi}{M}}^{\frac{\pi}{M}}} \middle| {{NTF}\left( ^{\omega} \right)} \middle| {}_{2}{\omega} \right.}\end{matrix}\quad} & \left\lbrack {{Equation}\quad 10} \right\rbrack\end{matrix}$

[0073] where, M is an oversampling ratio, σ_(e) is a standard deviationof quantization errors and Δ is a minimum quantization scale.

[0074] Substituting the equations 9 and 10 into the equation 8, theresult is:

[0075] Equation 11 $\begin{matrix}{{NSG} \equiv {10\log_{10}\frac{1}{\int_{- \frac{\pi}{M}}^{\frac{\pi}{M}}\left| {{NTF}\left( ^{\omega} \right)} \middle| {}_{2}{\omega} \right.}}} & \left\lbrack {{Equation}\quad 11} \right\rbrack\end{matrix}$

[0076] In the above equation 11, NTF(e^(iω)) is a Fourier transform ofNTF(z), which is defined as in the below equation 12.

NTF(e ^(iω))=|NTF(z)|_(z=e) _(^(iω))   Equation 12

[0077] For integration of the above equation 11, a more detaileddescription will hereinafter be given of an integrand in the right sideof the equation 11.

[0078] Substituting the equation 7 into the integrand in the equation11, the result is:

[0079] Equation 13 $\begin{matrix}{\begin{matrix}{\left| {N\quad T\quad {F\left( ^{\quad \omega} \right)}} \right|^{2} = \quad {\left( {{- 1} + {\sum\limits_{k = 1}^{N}{a_{k}z^{- k}}}} \right) \cdot \left( {{- 1} + {\sum\limits_{k = 0}^{N}{a_{k}z^{k}}}} \right)}} \\{= \quad \left( {{- 1} + {a_{1}e^{- {j\omega}}} + {a_{2}e^{- {j2\omega}}} +} \right.} \\{{\quad \left. {a_{3}e^{- {j3\omega}}\Lambda \quad a_{N}e^{{- j}\quad N\quad \omega}} \right)} \cdot \left( {{- 1} + {a_{1}e^{+ {j\omega}}} +} \right.} \\{\quad \left. {{a_{2}e^{+ {j2\omega}}} + {a_{3}e^{+ {j3\omega}}\Lambda \quad a_{N}e^{{- j}\quad N\quad \omega}}} \right)}\end{matrix}\quad} & \left\lbrack {{Equation}\quad 13} \right\rbrack\end{matrix}$

[0080] Expanding the above equation 13, respective terms can be arrangedabout cos Kω) (K=0, 1, 2, . . . , N) as in the following equation 14.

[0081] Equation 14 $\begin{matrix}{\left| {N\quad T\quad {F\left( ^{\quad \omega} \right)}} \right|^{2} = {\left( {1 + a_{1}^{2} + a_{2}^{2} + {a_{3}^{2}\Lambda \quad a_{N}^{2}}} \right) + {2\left( {{- a_{1}} + {a_{1}a_{2}} + {a_{2}a_{3}} + {\Lambda \quad a_{N - 1}a_{N}}} \right)\cos \quad \omega} + {2\left( {{- a_{2}} + {a_{1}a_{3}} + {a_{2}a_{4}} + {\Lambda \quad a_{N - 2}a_{N}}} \right)\cos \quad 2\omega} + {2\left( {{- a_{3}} + {a_{1}a_{4}} + {a_{2}a_{5}} + {\Lambda \quad a_{N - 3}a_{N}}} \right)\cos \quad 3\quad \omega} + \Lambda + {2\left( {{- a_{N - 1}} + {a_{1}a_{N}}} \right){\cos \left( {N - 1} \right)}\omega} + {2\left( {- a_{N}} \right)\cos \quad N\quad \omega}}} & \left\lbrack {{Equation}\quad 14} \right\rbrack\end{matrix}$

[0082] Substituting the above equation 14 into the integrand in theequation 11, the resulting integrand can be expressed as in the belowequation 15.

[0083] Equation 15 $\begin{matrix}{{\int_{- \frac{\pi}{M}}^{\frac{\pi}{M}}\left| {N\quad T\quad {F\left( ^{\quad \omega} \right)}} \middle| {}_{2}{\omega} \right.} = \quad {{{\left( {1 + a_{1}^{2} + a_{2}^{2} + {a_{3}^{2}\Lambda \quad a_{N}^{2}}} \right)\frac{2\pi}{M}} + {4\left( {{- a_{1}} + {a_{1}a_{2}} + {a_{2}a_{3}} + {\Lambda \quad a_{N - 1}a_{N}}} \right)\sin \frac{\pi}{M}} + {\frac{4}{2}\left( {{- a_{2}} + {a_{1}a_{3}} + {a_{2}a_{4}} + {\Lambda \quad a_{N - 2}a_{N}}} \right)\sin \frac{2\pi}{M}} + {\frac{4}{3}\left( {{- a_{3}} + {a_{1}a_{4}} + {a_{2}a_{5}} + {\Lambda \quad a_{N - 3}a_{N}}} \right)\sin \frac{2\pi}{M}} + \Lambda + {\frac{4}{\left( {N - 1} \right)}\left( {{- a_{N - 1}} + {a_{1}a_{N}}} \right)\sin \frac{\left( {N - 1} \right)\pi}{M}} + {\frac{4}{N}\left( {- a_{N}} \right)\sin \quad \frac{N\quad \pi}{M}}} \equiv {N\quad S\quad G^{*}}}} & \left\lbrack {{Equation}\quad 15} \right\rbrack\end{matrix}$

[0084] 2) Conditions of Real Coefficients for Optimization of ObjectiveFunction

[0085] The above equation 11 defining the objective function is in theform of a logarithmic function of a reciprocal of an integral function.In this connection, a condition where the objective function becomes amaximal value is equal to a condition where the integral functionthereof becomes a minimal value.

[0086] The objective function of the equation 11 is a function ofseveral variables for real coefficients a_(k) and the integral functionthereof is a function of several variables for the real coefficientsa_(k), too. As seen from the above equation 15, the integral functionincreases infinitely positively with regard to the respective realcoefficients a_(k). In this connection, provided that there is oneextreme value at which all differential quotients for the respectivereal coefficients are 0, the integral function will become the minimalvalue and the objective function will become the maximal value. That theobjective function becomes the maximal value means that the noiseshaping gain becomes the maximal value. Accordingly, the extreme value,or the minimal value, is obtained by defining the integral function ofthe objective function as the auxiliary function NSG* as in the equation15 and partially differentiating it.

[0087] The following equation 16 expresses a condition where theauxiliary function NSG* is partially differentiated with regard to therespective real coefficients and the resulting partial differentialquotients become ‘0’.

[0088] Equation 16 $\begin{matrix}{\frac{\partial\left( {N\quad S\quad G^{*}} \right)}{\partial a_{k}} = {0\quad \left( {{w\quad h\quad e\quad r\quad e},{\kappa = 1},2,3,\Lambda,N} \right)}} & \left\lbrack {{Equation}\quad 16} \right\rbrack\end{matrix}$

[0089] Substituting the equation 15 defining the auxiliary function NSG*into the above equation 16 and expanding the resulting equation withrespect to κ=1, the result is:

[0090] Equation 17 $\begin{matrix}{\begin{matrix}{\frac{\partial\left( {NSG}^{*} \right)}{\partial a_{1}} = \quad {{\frac{4\pi}{M}a_{1}} - {4\sin \frac{\pi}{M}} + {4\left( {\sin \frac{\pi}{M}} \right)a_{2}} +}} \\{\quad {{\frac{4}{2}\left( {\sin 2\frac{\pi}{M}} \right)a_{3}} + {\Lambda \frac{4}{\left( {N - 1} \right)}\left( {{\sin \left( {N - 1} \right)}\frac{\pi}{M}} \right)a_{N}}}} \\{= \quad 0}\end{matrix}\quad} & \left\lbrack {{Equation}\quad 17} \right\rbrack\end{matrix}$

[0091] Expanding the resulting equation with respect to κ=2, the resultis:

[0092] Equation 18 $\begin{matrix}{\begin{matrix}{\frac{\partial\left( {NSG}^{*} \right)}{\partial a_{2}} = \quad {{\frac{4\pi}{M}a_{2}} + {4\left( {\sin \frac{\pi}{M}} \right)a_{1}} - {\frac{4}{2}\sin \frac{2\pi}{M}} +}} \\{\quad {{\frac{4}{2}\left( {\sin \frac{2\pi}{M}} \right)a_{4}} + \Lambda + \frac{4}{\left( {N - 2} \right)}}} \\{\quad {\left( {\sin \frac{\left( {N - 2} \right)\pi}{M}} \right)a_{N}}} \\{= \quad 0}\end{matrix}\quad} & \left. \left\lbrack {{Equation}\quad 18} \right. \right)\end{matrix}$

[0093] N condition equations generated by expanding the resultingequation up to κ=N can be simply expressed by the following equation 19,which is a determinant.

GA=B   Equation 19

[0094] where, A is an order-N column vector of the real coefficients, Gis an N×N matrix of constants and B is an order-N column vector ofdifferential quotients of the auxiliary function NSG* with regard to therespective real coefficients.

[0095] A and B in the above equation 19 can be expressed by thefollowing equations 20 and 21, respectively.

[0096] Equation 20 $\begin{matrix}{A = \begin{pmatrix}a_{1} \\a_{2} \\a_{3} \\M \\a_{N}\end{pmatrix}} & \left\lbrack {{Equation}\quad 20} \right\rbrack\end{matrix}$

[0097] Equation 21 $\begin{matrix}{B = \begin{pmatrix}b_{1} \\b_{2} \\b_{3} \\M \\b_{N}\end{pmatrix}} & \left\lbrack {{Equation}\quad 21} \right\rbrack\end{matrix}$

[0098] Respective terms b,(i=0˜N) of the order-N column vector B can begiven as in the below equation 22.

[0099] Equation 22 $\begin{matrix}{b_{i} = {\frac{4}{i}\sin i\quad \frac{\pi}{M}}} & \left\lbrack {{Equation}\quad 22} \right\rbrack\end{matrix}$

[0100] The matrix G can be expressed by the following equation 23 andrespective elements thereof can be simply expressed as in the belowequation 24.

[0101] Equation 23 $\begin{matrix}\begin{matrix}{G = \left\{ g_{ij} \right\}} \\{= \begin{bmatrix}\frac{4\pi}{M} & {\frac{4}{1}\sin \frac{\pi}{M}} & {\frac{4}{2}\sin \frac{2\pi}{M}} & {\Lambda \frac{4}{N - 1}{\sin \left( {N - 1} \right)}\frac{\pi}{M}} \\{\frac{4}{1}\sin \frac{\pi}{M}} & \frac{4\pi}{M} & {\frac{4}{1}\sin \frac{\pi}{M}} & {\Lambda \frac{4}{N - 2}{\sin \left( {N - 2} \right)}\frac{\pi}{M}} \\M & M & M & M \\{\frac{4}{N - 1}{\sin \left( {N - 1} \right)}\frac{\pi}{M}} & {\frac{4}{N - 2}{\sin \left( {N - 2} \right)}\frac{\pi}{M}} & \Lambda & \frac{4\pi}{M}\end{bmatrix}}\end{matrix} & \left\lbrack {{Equation}\quad 23} \right\rbrack\end{matrix}$

[0102] Equation 24 $\begin{matrix}{g_{ij} = \left\{ {\begin{matrix}{\quad \frac{4\pi}{M}} & {\quad \left( {i = j} \right)} \\{\quad {\frac{4}{\left( {j - i} \right)}{\sin \left( {j - i} \right)}\frac{\pi}{M}}} & {\quad \left( {{i \neq j},{j > i}} \right)} \\{\quad g_{ij}} & {\quad \left( {{i \neq j},{j < i}} \right)}\end{matrix}\quad} \right.} & \left\lbrack {{Equation}\quad 24} \right\rbrack\end{matrix}$

[0103] where, M is an oversampling ratio.

[0104] The below equation 25 can be obtained by obtaining an inversematrix of the matrix G and multiplying both sides of the equation 19 bythe obtained inverse matrix. Then, all optimum real coefficients a_(κ)can be obtained by arranging the right side of the equation 25.

A=G⁻¹B   Equation 25

[0105] By defining the objective function and obtaining the conditionsof real coefficients capable of optimizing the objective function, asstated above, the optimum real coefficients can be calculated withrespect to any oversampling ratio or any order of the noise transferfunction, thereby making the optimum noise-shaping filter.

[0106] 2.1) Optimum Real Coefficients for 8-Times Oversampling

[0107] As stated previously, all coefficients of a noise transferfunction of a conventional noise-shaping filter are integers as shown inthe table 1. The below table 2 shows optimum real coefficients ofsecond-order to seventh-order noise transfer functions obtained for8-times oversampling by a method for making a noise-shaping filteraccording to the present invention. In the present method, the equations22, 24 and 25 are programmed in a typical calculator, for example, acomputer to input a predetermined oversampling ratio M of thenoise-shaping filter and a predetermined order N of the noise transferfunction and automatically calculate the optimum real coefficients onthe basis of the inputted oversampling ratio M and order N. This programor programming technique is implemented in a typical manner. TABLE 2ORDER 2nd 3rd 4th 5th 6th 7th α₁ 1.9290 2.8891 3.8502 4.8116 5.77326.7350 α₂ −0.9795 −2.8703 −5.6846 −9.4235 −14.0874 −19.6767 α₃ 0.98023.8129 9.3872 18.5927 32.3201 α₄ −0.9805 −4.7559 −13.9971 −32.2316 α₅0.9806 5.6991 19.5151 α₆ −0.9807 −6.6427 α₇ 0.9807

[0108] Although coefficient values are shown in the above table 2 toonly four decimal places on account of limited space, those skilled inthe art will appreciate that the coefficient values can be calculated inmore detail from the above-described equations.

[0109] FIGS. 6 to 11 show noise transfer characteristics-by-orders ofthe optimum noise-shaping filter implemented for the 8-timesoversampling on the basis of the coeffients of the table 2. In eachdrawing, the abscissa represents a frequency normalized into a digitalsignal, and the ordinate represents the level of suppressed noise. Thelevel 0 dB on the ordinate represents noise, not suppressed at all. Thelower level on the ordinate indicates better performance because alarger amount of noise is suppressed. A dotted line indicatescharacteristics of a conventional noise-shaping filter and a solid lineindicates characteristics of a proposed noise-shaping filter. A one-dotchain line so overlapping the solid line as to be difficult todistinguish it from the solid line indicates characteristics of anoise-shaping filter having optimum real coefficients whose decimalplaces are approximated by a 16-bit binary number, as will be describedin “3. APPROXIMATION OF OPTIMUM REAL COEFFICIENTS OF NOISE TRANSFERFUNCTION”. The noise transfer function of the noise-shaping filterapproximating the optimum value so closely overlaps the optimum value asto be indistinguishable from it.

[0110]FIG. 6 is a graph showing a second-order noise transfer functionof the proposed noise-shaping filter, expressible by the equation 6 or7. In this drawing, the oversampling ratio is 8 times. On the abscissaof the graph, an area below 8/π≈0.3927 is an audio frequency band or afrequency band of interest, where the smaller the value of the noisetransfer function becomes, the better the noise suppression performancebecomes. It can be seen from a curved line in FIG. 6 that the optimizednoise-shaping filter and the noise-shaping filter approximating it havesmaller noise transfer function values than that of the conventionalnoise-shaping filter on the whole. Although the optimized noise-shapingfilter has a greater noise transfer function value than that of theconventional noise-shaping filter in a part of the frequency band ofinterest, it has a greater noise suppression gain corresponding to theoverall result than that of the conventional noise-shaping filter, asshown in the below table 3. The optimized noise-shaping filter furtherhas a smaller noise transfer function value than that of theconventional noise-shaping filter even in an area beyond the audiofrequency band or frequency band of interest.

[0111] FIGS. 7 to 11 are graphs showing third-order to seventh-ordernoise transfer functions of the proposed noise-shaping filter, whichhave substantially the same characteristics as those of FIG. 6 and adetailed description thereof will thus be omitted. In conclusion, it canbe seen from the respective drawings that the proposed noise-shapingfilter has significantly improved characteristics at the audio frequencyband.

[0112] The below table 3 shows a comparison between the conventionalnoise-shaping filter and the digital noise-shaping filter with realcoefficients according to the present invention with regard to the noiseshaping gain defined in the equation 8. TABLE 3 2nd 3rd 4th 5th 6th 7thCONVENTIONAL FILTER 24.4 34.0 43.2 52.3 61.2 70.0 (1) PROPOSED OPTIMUM27.9 42.0 56.1 70.3 84.5 98.6 FILTER (2) (2)-(1) 3.6 8.0 12.9 18.0 23.328.6

[0113]FIG. 12 is a graph showing the results of the above table 3. Inthis drawing, a symbol ⋄ indicates the noise shaping gain of theconventional noise-shaping filter and a symbol ◯ indicates the noiseshaping gain of the optimum noise-shaping filter. A curved line markedwith “+” signs indicates a noise shaping gain of a noise-shaping filterhaving real coefficients whose decimal places are approximated by a16-bit binary number, as will be described in “3. APPROXIMATION OFOPTIMUM REAL COEFFICIENTS OF NOISE TRANSFER FUNCTION”. It is impossibleto distinguish a curved line indicative of an optimum noise shaping gainand a curved line indicative of an approximate noise shaping gain fromeach other because they nearly overlap each other.

[0114] As seen from the graph of FIG. 12, the digital noise-shapingfilter with the optimum real coefficients according to the presentinvention has excellent noise shaping gain and noise suppressioncharacteristics over the conventional noise-shaping filter. It can alsobe seen from this drawing that, as the noise-shaping filter becomeslower in order, it more simply approximates the optimized noise-shapingfilter and has a smaller performance difference from the optimizednoise-shaping filter.

[0115] 3. Approximation of Optimum Real Coefficients of Noise TransferFunction

[0116] 1) Approximation of Real Coefficients of Second-OrderNoise-Shaping Filter

[0117] The present invention provides an approximation method capable ofimplementing an optimum real-coefficient noise-shaping filter which hasnoise suppression performance approximating that of the optimumnoise-shaping filter and is higher in calculation speed than the optimumnoise-shaping filter.

[0118]FIG. 13 is a graph showing a noise shaping gain of a second-ordernoise-shaping filter whose real coefficients are approximated to onlyfour decimal places by a binary number. In this drawing, a dotted lineindicates a noise transfer function of a conventional noise-shapingfilter and a solid line indicates a noise transfer function of anoptimized noise-shaping filter. A one-dot chain line indicates a noisetransfer function of a noise-shaping filter having real coefficientswhose four decimal places are approximated by a binary number. Fromcomparison between FIG. 13 and FIG. 6, it can be seen that thenoise-shaping filter having the noise transfer function with theapproximated real coefficients is implemented more simply than thenoise-shaping filter having the noise transfer function with theoptimized real coefficients and has characteristics almost analogous tothose of the noise-shaping filter with the optimized real coefficients.The approximated real coefficients of the noise-shaping filter can beexpressed by the below equations 26 and 27, respectively.

[0119] Equation 26 $\begin{matrix}{a_{1} = {1.9290 = {{2 - \left( {1 - 0.9290} \right)} = {{2 - 0.0710} = {\left. {10_{(2)} - {0.000100\Lambda_{(2)}}}\rightarrow a_{1} \right. = {{10_{(2)} - 0.0001_{(2)}} = {{2 - \frac{1}{16}} = 1.9375}}}}}}} & \left\lbrack {{Equation}\quad 26} \right\rbrack\end{matrix}$

a₂=−1.0   Equation 27

[0120] As seen from the above equation 26, each real coefficient can beapproximated by binary decimals of 4 bits by (1) dividing the originalreal coefficient into a larger integer (2 in the equation 26) than theoriginal real coefficient and decimals (0.0710 in the equation 26), (2)expressing the integer and decimals respectively as binary numbers (inthe equation 26, the binary integer is 10₍₂₎ and the binary decimals are0.000100 . . . ₍₂₎), (3) taking only 4 bits of the binary decimals(0.0001₍₂₎ in the equation 26) and (4) expressing a value approximatedby the binary decimals of 4 bits as a decimal number (1.9375 in theequation 26).

[0121] The noise-shaping filter can obtain a coefficient a₁ in theequation 26 by performing only one-time shifting and addition by anadder or subtracter without using a multiplier.

[0122]FIG. 14 is a conceptual diagram of an apparatus for approximatingan optimum real coefficient of the first term of a second-ordernoise-shaping filter for 8-times oversampling. It is common practicethat an adder or subtracter is simpler in construction and higher incalculation speed than a multiplier.

[0123] The multiplication of a coefficient a₂ can be more simplyperformed by changing only a sign. Calculating a noise shaping gain onthe basis of such a simple calculation, the result is:

[0124] Equation 28 $\begin{matrix}{{N\quad S\quad G} = {{27.6\left\lbrack {d\quad B} \right\rbrack}\quad \left( {{a_{1} = {1.9375 = {2 - \frac{1}{16}}}},{a_{2} = {- 1.0}}} \right)}} & \left\lbrack {{Equation}\quad 28} \right\rbrack\end{matrix}$

[0125] This noise shaping gain is improved by 3.3 dB over theconventional second-order noise-shaping filter (NSG=24.3) shown in thetable 3, and has a fine difference of only 0.3 dB from that of theoptimized second-order filter (NSG=27.9).

[0126] It should be noted herein that the method for approximating realcoefficients of the noise transfer function of the noise-shaping filteraccording to the present invention is also applicable to a third-orderor higher-order noise-shaping filter.

[0127] 2) Approximation of Real Coefficients of Third-OrderNoise-Shaping filter

[0128] For a third-order noise-shaping filter, real coefficients can beapproximated appropriately in the same manner as those of thesecond-order noise-shaping filter. By expressing ideal coefficients a₁and a₂ to only four decimal places as binary numbers, the coefficientsa₁ and a₂ can be approximated as in the below equations 29 and 30,respectively.

[0129] Equation 29 $\begin{matrix}{a_{1} = {\left. {- 2.8891}\rightarrow{a_{1} \approx 2.8750} \right. = {3 - \frac{1}{8}}}} & \left\lbrack {{Equation}\quad 29} \right\rbrack\end{matrix}$

[0130] Equation 30 $\begin{matrix}{a_{2} = {\left. {- 2.8703}\rightarrow{a_{2} \approx {- 2.8750}} \right. = {- \left( {3 - \frac{1}{8}} \right)}}} & \left\lbrack {{Equation}\quad 30} \right\rbrack\end{matrix}$

[0131] It can be seen from the above approximate values that thecoefficient multiplication can be carried out by a simple and rapidhardware configuration in a similar manner to the second-ordernoise-shaping filter. The below table 4 shows a noise shaping gain of aconventional third-order noise-shaping filter, a noise shaping gain ofan optimized third-order noise-shaping filter and a noise shaping gainof a third-order noise-shaping filter having real coefficients whosedecimal places are approximated by 4 bits. As seen from FIG. 13,however, a fourth-order or higher-order noise-shaping filter having realcoefficients whose decimal places are approximated by 4 bits has asmaller noise shaping gain than that of a conventional fourth-order orhigher-order noise-shaping filter. In this regard, for 4-bitapproximation, it is preferred that the order of the noise-shapingfilter is three or less. TABLE 4 DIFFERENCE NOISE FROM SHAPING OPTIMUMFILTER TYPE GAIN IMPROVED GAIN VALUE CONVENTIONAL 34.0 dB OPTIMUM 42.0dB 8.0 dB APPROXIMATION 39.6 dB 5.6 dB −2.4 dB OF DECIMAL PLACES BY 4BITS

[0132] 3) Approximation of Real Coefficients of Nth-Order Noise-ShapingFilter

[0133] Although the above approximations have been disclosed forillustrative purposes with respect to the second-order and third-ordernoise-shaping filters, it should be noted herein that the method forapproximating real coefficients of the noise transfer function of thenoise-shaping filter according to the present invention is not limitedto only the second-order and third-order noise-shaping filters. Thoseskilled in the art will readily appreciate from an item about a filterapproximating an optimum filter (approximation by binary 4 bits) in thebelow table 5 that the method for approximating real coefficients of thenoise transfer function of the noise-shaping filter according to thepresent invention is also applicable to noise-shaping filters of allorders such as order 4, order 5, order 6, order 7, etc.

[0134] 4) Range of Bits for Binary Approximation

[0135] The approximation by binary 4 bits has been described as anexample. It should be noted herein that this invention is not limited tothe 4-bit approximation and is applicable to any number of bits forbinary approximation, such as more than 4 bits, for example, 8 bits, 16bits, etc.

[0136] The below table 5 shows noise shaping gains of conventionalsecond-order to seventh-order noise-shaping filters, noise shaping gainsof optimum second-order to seventh-order noise-shaping filters, andnoise shaping gains of second-order to seventh-order noise-shapingfilters employing approximations by 2 bits, 3 bits, 4 bits and 16 bits.FIG. 15 is a graph showing a noise shaping gain of a second-ordernoise-shaping filter whose real coefficients are approximated to twodecimal places by a binary number, and FIG. 16 is a graph showing anoise shaping gain of a second-order noise-shaping filter whose realcoefficients are approximated to three decimal places by a binarynumber. From an item about a filter approximating an optimum filter(approximation by binary 2 bits) in the below table 5, an item about afilter approximating an optimum filter (approximation by binary 3 bits)in the table 5, and FIGS. 15 and 16, it can be seen that, forapproximations of decimals by 2 bits and 3 bits, noise shaping gains aresmaller than or almost equal to those of conventional noise-shapingfilters. In other words, for real coefficients whose decimal places areapproximated by 3 bits or less, there is no meaning in approximation.Thus, it is preferred to approximate real coefficients of noise-shapingfilters by 4 bits or more. TABLE 5 2nd 3rd 4th 5th 6th 7th CONVENTIONALFILTER (1) 24.4 34.0 43.2 52.3 61.2 70.0 PROPOSED OPTIMUM FILTER (2)27.9 42.0 56.1 70.3 84.5 98.6 FILTER APPROXIMATING OPTIMUM FILTER *APPROXIMATION BY BINARY 2 BITS 24.4 13.1 13.4 26.5 61.1 13.5 FILTERAPPROXIMATING OPTIMUM FILTER * APPROXIMATION BY BINARY 3 BITS 22.3 39.619.0 33.0 32.1 20.0 FILTER APPROXIMATING OPTIMUM FILTER * APPROXIMATIONBY BINARY 4 BITS 27.6 39.6 37.8 39.1 25.9 36.8 FILTER APPROXIMATINGOPTIMUM FILTER (3) * APPROXIMATION BY BINARY 16 BITS 27.9 42.0 56.1 70.384.3 98.6 (2)-(1) 3.6 8.0 12.9 18.0 23.3 28.6 (3)-(1) 3.6 8.0 12.9 18.023.1 28.6

[0137] The lower part of the above table 5 shows a difference between anoise shaping gain of a conventional noise-shaping filter and a noiseshaping gain of an optimum noise-shaping filter and a difference betweenthe noise shaping gain of the conventional noise-shaping filter and anoise shaping gain of a noise-shaping filter approximating the optimumnoise-shaping filter (approximation by binary 16 bits). It can be seenfrom such differences that the optimum noise-shaping filter exhibits adifference of 3 dB or more in all orders, which is a sufficientlysignificant difference for a human being to easily detect.

[0138] Industrial Applicability

[0139] As apparent from the above description, the present inventionprovides a digital noise-shaping filter and a method for making thesame, wherein respective coefficients of a noise transfer function ofthe noise-shaping filter are not simple integers, but appropriate realcoefficients. Further, the digital noise-shaping filter is able tomaximize noise suppression performance.

[0140] Further, according to the present invention, the digitalnoise-shaping filter shows excellent noise suppression performance overthose of conventional noise-shaping filters in the same order.Therefore, this noise-shaping filter is stable in operation andexcellent in noise suppression performance.

[0141] Further, according to this invention, the digital noise-shapingfilter is more stable in operation than conventional noise-shapingfilters under the condition that the same noise suppression performanceis obtained.

[0142] Further, according to this invention, the digital noise-shapingfilter has real coefficients approximating optimum values so that it canhave almost the same performance as that of an optimum noise-shapingfilter without increasing a calculation complexity.

[0143] Further, according to this invention, the digital noise-shapingfilter has a lowest-order noise transfer function under the conditionthat the same noise suppression performance is obtained, resulting in anincrease in system stability.

[0144] Although the preferred embodiments of the present invention havebeen disclosed for illustrative purposes, those skilled in the art willappreciate that various modifications, additions and substitutions arepossible, without departing from the scope and spirit of the inventionas disclosed in the accompanying claims.

1. A digital noise-shaping filter for a delta-sigma data converter,comprising a noise transfer function expressed by the following equation1, said noise transfer function having real coefficients: NTF(z)=−1+a ₁z ⁻¹ +a ₂ z ⁻² +Λ+a _(N) z ^(−N)   Equation 1 where, NTF(z) is az-transform of said noise transfer function, a₁, a₂, . . . a_(N) aresaid real coefficients of said noise transfer function, and N is anorder of said noise-shaping filter.
 2. The digital noise-shaping filteras set forth in claim 1, wherein said real coefficients of said noisetransfer function are obtained by: 1) defining an objective functionenabling a quantitative evaluation of the performance of saidnoise-shaping filter; 2) obtaining real coefficient conditions capableof optimizing said objective function; and 3) mathematically calculatingoptimum real coefficients satisfying said real coefficient conditions.3. The digital noise-shaping filter as set forth in claim 2, whereinsaid objective function is defined as the ratio of the amount of energyof quantization noise before being shaped to the amount of energy ofquantization noise after being shaped, at a specific frequency band. 4.The digital noise-shaping filter as set forth in claim 2, wherein saidobjective function is expressed by the following equation 2: Equation 2${N\quad S\quad G} \equiv {10\log_{10}\frac{P_{r\quad q}}{P_{n\quad s}}}$

where, NSG is said objective function, P_(ns) is power of energye_(ns)(n) of quantization noise before shaped, and P_(rq) is power ofenergy e_(rq)(n) of quantization noise after being shaped.
 5. Thedigital noise-shaping filter as set forth in claim 4, wherein adenominator in a logarithm of said objective function is expressed bythe following equation 3 by: 1) arranging said power P_(ns) of theenergy e_(ns)(n) of the quantization noise before being shaped and saidpower P_(rq) of the energy e_(rq)(n) of the quantization noise afterbeing shaped, on the basis of an oversampling ratio M and a minimumquantization scale Δ; and 2) substituting said equation 2 into thearranged objective function: Equation 3 $\begin{matrix}{{\int_{- \frac{\pi}{M}}^{\frac{\pi}{M}}\left| {N\quad T\quad {F\left( ^{\quad \omega} \right)}} \middle| {}_{2}{\omega} \right.} = \quad {{{\left( {1 + a_{1}^{2} + a_{2}^{2} + {a_{3}^{2}\Lambda \quad a_{N}^{2}}} \right)2\frac{\pi}{M}} + {4\left( {{- a_{1}} + {a_{1}a_{2}} + {a_{2}a_{3}} + {\Lambda \quad a_{N - 1}a_{N}}} \right)\sin \frac{\pi}{M}} + {\frac{4}{2}\left( {{- a_{2}} + {a_{1}a_{3}} + {a_{2}a_{4}} + {\Lambda \quad a_{N - 2}a_{N}}} \right)\sin 2\frac{\pi}{M}} + {\frac{4}{3}\left( {{- a_{3}} + {a_{1}a_{4}} + {a_{2}a_{5}} + {\Lambda \quad a_{N - 3}a_{N}}} \right)\sin 3\frac{\pi}{M}} + \Lambda + {\frac{4}{\left( {N - 1} \right)}\left( {{- a_{N - 1}} + {a_{1}a_{N}}} \right){\sin \left( {N - 1} \right)}\frac{\pi}{M}} + {\frac{4}{N}\left( {- a_{N}} \right)\sin N\quad \frac{\pi}{M}}} \equiv {N\quad S\quad G^{*}}}} & \left\lbrack {{Equation}\quad 3} \right\rbrack\end{matrix}$


6. The digital noise-shaping filter as set forth in claim 5, whereinsaid real coefficient conditions capable of optimizing said objectivefunction are obtained by partially differentiating said equation 3 ofsaid logarithm denominator of said objective function with respect to areal coefficient of each term and setting each real coefficientcorresponding to the resulting extreme value as each real coefficient ofsaid equation
 1. 7. The digital noise-shaping filter as set forth inclaim 6, wherein said optimum real coefficients satisfying said realcoefficient conditions are elements of a matrix A expressed by thefollowing equation 4: A=G⁻¹B   Equation 4 where, G is an order-N squarematrix expressed by the following equation 5 and B is an order-N columnvector whose elements are expressed by the following equation 6:Equation 5 $\begin{matrix}{\begin{matrix}{G = \left\{ g_{ij} \right\}} \\{= \begin{bmatrix}\frac{4\pi}{M} & {\frac{4}{1}\sin \frac{\pi}{M}} & {\frac{4}{2}\sin \frac{\pi}{M}} & {\Lambda \frac{4}{N - 1}\sin \frac{\left( {N - 1} \right)\pi}{M}} \\{\frac{4}{1}\sin \frac{\pi}{M}} & \frac{4\pi}{M} & {\frac{4}{1}\sin \frac{\pi}{M}} & {\Lambda \frac{4}{N - 2}\sin \frac{\left( {N - 2} \right)\pi}{M}} \\M & M & M & M \\{\frac{4}{N - 1}\sin \frac{\left( {N - 1} \right)\pi}{M}} & {\frac{4}{N - 2}\sin \frac{\left( {N - 2} \right)\pi}{M}} & \Lambda & \frac{4\pi}{M}\end{bmatrix}}\end{matrix}\quad} & \left\lbrack {{Equation}\quad 5} \right\rbrack\end{matrix}$

Equation 6 $\begin{matrix}{b_{i} = {\frac{4}{i}\sin i\quad \frac{\pi}{M}}} & \left\lbrack {{Equation}\quad 6} \right\rbrack\end{matrix}$


8. The digital noise-shaping filter as set forth in claim 2 or claim 7,wherein said real coefficients of said noise transfer function arevalues approximating said optimum real coefficients.
 9. The digitalnoise-shaping filter as set forth in claim 8, wherein said approximatereal coefficients are values approximated by a binary number.
 10. Thedigital noise-shaping filter as set forth in claim 9, wherein saidvalues approximated by said binary number are values approximated tofour or more decimal places.
 11. The digital noise-shaping filter as setforth in claim 10, wherein said values approximated by said binarynumber are said values approximated to four decimal places when theorder of said noise-shaping filter is 2 or
 3. 12. The digitalnoise-shaping filter as set forth in claim 11, wherein, for calculationof said noise transfer function of said equation 1, said valuesapproximated to four or more decimal places are calculated by means ofan adder or subtracter.
 13. A method for making a digital noise-shapingfilter for a delta-sigma data converter, said digital noise-shapingfilter comprising a noise transfer function expressed byNTF(z)=−1+a₁z⁻¹+a₂z⁻²+Λ+a_(N)z^(−N), said noise transfer function havingreal coefficients a₁, a₂, . . . a_(N), wherein said real coefficients ofsaid noise transfer function are obtained by the steps of: a) definingan objective function enabling a quantitative evaluation of theperformance of said noise-shaping filter; b) obtaining real coefficientconditions capable of optimizing said objective function; and c)mathematically calculating optimum real coefficients satisfying saidreal coefficient conditions.
 14. The method as set forth in claim 13,wherein said objective function is expressed by the following equation7: Equation 7 $\begin{matrix}{{N\quad S\quad G} \equiv {10\log_{10}\frac{P_{r\quad q}}{P_{n\quad s}}}} & \left\lbrack {{Equation}\quad 7} \right\rbrack\end{matrix}$

where, NSG is said objective function, P_(ns) is power of energye_(ns)(n) of quantization noise before being shaped, and P_(rq) is powerof energy e_(rq)(n) of quantization noise after being shaped.
 15. Themethod as set forth in claim 14, wherein a denominator in a logarithm ofsaid objective function is expressed by the following equation 8 by: 1)arranging said power P_(ns) of the energy e_(ns)(n) of the quantizationnoise before being shaped and said power P_(rq) of the energy e_(rq)(n)of the quantization noise after being shaped, on the basis of anoversampling ratio M and a minimum quantization scale Δ; and 2)substituting said equation 7 into the arranged objective function:Equation 8 $\begin{matrix}{{\int_{- \frac{\pi}{M}}^{\frac{\pi}{M}}\left| {N\quad T\quad {F\left( ^{\quad \omega} \right)}} \middle| {}_{2}{\omega} \right.} = \quad {{{\left( {1 + a_{1}^{2} + a_{2}^{2} + {a_{3}^{2}\Lambda \quad a_{N}^{2}}} \right)2\frac{\pi}{M}} + {4\left( {{- a_{1}} + {a_{1}a_{2}} + {a_{2}a_{3}} + {\Lambda \quad a_{N - 1}a_{N}}} \right)\sin \frac{\pi}{M}} + {\frac{4}{2}\left( {{- a_{2}} + {a_{1}a_{3}} + {a_{2}a_{4}} + {\Lambda \quad a_{N - 2}a_{N}}} \right)\sin 2\frac{\pi}{M}} + {\frac{4}{3}\left( {{- a_{3}} + {a_{1}a_{4}} + {a_{2}a_{5}} + {\Lambda \quad a_{N - 3}a_{N}}} \right)\sin 3\frac{\pi}{M}} + \Lambda + {\frac{4}{\left( {N - 1} \right)}\left( {{- a_{N - 1}} + {a_{1}a_{N}}} \right){\sin \left( {N - 1} \right)}\frac{\pi}{M}} + {\frac{4}{N}\left( {- a_{N}} \right)\sin N\quad \frac{\pi}{M}}} \equiv {N\quad S\quad G^{*}}}} & \left\lbrack {{Equation}\quad 8} \right\rbrack\end{matrix}$


16. The method as set forth in claim 15, wherein said real coefficientconditions capable of optimizing said objective function are obtained bypartially differentiating said equation 8 of said logarithm denominatorof said objective function with respect to a real coefficient of eachterm and setting each real coefficient corresponding to the resultingextreme value as each real coefficient of said noise transfer function.17. The method as set forth in claim 16, wherein said optimum realcoefficients satisfying said real coefficient conditions are elements ofa matrix A expressed by the following equation 9: A=G⁻¹B   Equation 9where, G is an order-N square matrix expressed by the following equation10 and B is an order-N column vector whose elements are expressed by thefollowing equation 11: Equation 10 $\begin{matrix}{\begin{matrix}{G = \left\{ g_{ij} \right\}} \\{= \begin{bmatrix}\frac{4\pi}{M} & {\frac{4}{1}\sin \frac{\pi}{M}} & {\frac{4}{2}\sin \frac{\pi}{M}} & {\Lambda \frac{4}{N - 1}\sin \frac{\left( {N - 1} \right)\pi}{M}} \\{\frac{4}{1}\sin \frac{\pi}{M}} & \frac{4\pi}{M} & {\frac{4}{1}\sin \frac{\pi}{M}} & {\Lambda \frac{4}{N - 2}\sin \frac{\left( {N - 2} \right)\pi}{M}} \\M & M & M & M \\{\frac{4}{N - 1}\sin \frac{\left( {N - 1} \right)\pi}{M}} & {\frac{4}{N - 2}\sin \frac{\left( {N - 2} \right)\pi}{M}} & \Lambda & \frac{4\pi}{M}\end{bmatrix}}\end{matrix}\quad} & \left\lbrack {{Equation}\quad 10} \right\rbrack\end{matrix}$

Equation 11 $\begin{matrix}{b_{i} = {\frac{4}{i}\sin i\quad \frac{\pi}{M}}} & \left\lbrack {{Equation}\quad 11} \right\rbrack\end{matrix}$


18. The method as set forth in claim 13 or claim 17, wherein said realcoefficients of said noise transfer function are values approximatingsaid optimum real coefficients.
 19. The method as set forth in claim 18,wherein said approximate real coefficients are values approximated tofour or more decimal places by a binary number.
 20. The method as setforth in claim 19, wherein said values approximated by said binarynumber are said values approximated to four decimal places when theorder of said noise-shaping filter is 2 or
 3. 21. The method as setforth in claim 20, wherein, for calculation of said noise transferfunction, said values approximated to four or more decimal places by thebinary number are calculated by means of an adder or subtracter.